1 /* intprops.h -- properties of integer types 2 3 Copyright (C) 2001-2018 Free Software Foundation, Inc. 4 5 This program is free software: you can redistribute it and/or modify it 6 under the terms of the GNU Lesser General Public License as published 7 by the Free Software Foundation; either version 2.1 of the License, or 8 (at your option) any later version. 9 10 This program is distributed in the hope that it will be useful, 11 but WITHOUT ANY WARRANTY; without even the implied warranty of 12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 13 GNU Lesser General Public License for more details. 14 15 You should have received a copy of the GNU Lesser General Public License 16 along with this program. If not, see <https://www.gnu.org/licenses/>. */ 17 18 /* Written by Paul Eggert. */ 19 20 #ifndef _GL_INTPROPS_H 21 #define _GL_INTPROPS_H 22 23 #include <limits.h> 24 25 /* Return a value with the common real type of E and V and the value of V. */ 26 #define _GL_INT_CONVERT(e, v) (0 * (e) + (v)) 27 28 /* Act like _GL_INT_CONVERT (E, -V) but work around a bug in IRIX 6.5 cc; see 29 <https://lists.gnu.org/r/bug-gnulib/2011-05/msg00406.html>. */ 30 #define _GL_INT_NEGATE_CONVERT(e, v) (0 * (e) - (v)) 31 32 /* The extra casts in the following macros work around compiler bugs, 33 e.g., in Cray C 5.0.3.0. */ 34 35 /* True if the arithmetic type T is an integer type. bool counts as 36 an integer. */ 37 #define TYPE_IS_INTEGER(t) ((t) 1.5 == 1) 38 39 /* True if the real type T is signed. */ 40 #define TYPE_SIGNED(t) (! ((t) 0 < (t) -1)) 41 42 /* Return 1 if the real expression E, after promotion, has a 43 signed or floating type. */ 44 #define EXPR_SIGNED(e) (_GL_INT_NEGATE_CONVERT (e, 1) < 0) 45 46 47 /* Minimum and maximum values for integer types and expressions. */ 48 49 /* The width in bits of the integer type or expression T. 50 Padding bits are not supported; this is checked at compile-time below. */ 51 #define TYPE_WIDTH(t) (sizeof (t) * CHAR_BIT) 52 53 /* The maximum and minimum values for the integer type T. */ 54 #define TYPE_MINIMUM(t) ((t) ~ TYPE_MAXIMUM (t)) 55 #define TYPE_MAXIMUM(t) \ 56 ((t) (! TYPE_SIGNED (t) \ 57 ? (t) -1 \ 58 : ((((t) 1 << (TYPE_WIDTH (t) - 2)) - 1) * 2 + 1))) 59 60 /* The maximum and minimum values for the type of the expression E, 61 after integer promotion. E should not have side effects. */ 62 #define _GL_INT_MINIMUM(e) \ 63 (EXPR_SIGNED (e) \ 64 ? ~ _GL_SIGNED_INT_MAXIMUM (e) \ 65 : _GL_INT_CONVERT (e, 0)) 66 #define _GL_INT_MAXIMUM(e) \ 67 (EXPR_SIGNED (e) \ 68 ? _GL_SIGNED_INT_MAXIMUM (e) \ 69 : _GL_INT_NEGATE_CONVERT (e, 1)) 70 #define _GL_SIGNED_INT_MAXIMUM(e) \ 71 (((_GL_INT_CONVERT (e, 1) << (TYPE_WIDTH ((e) + 0) - 2)) - 1) * 2 + 1) 72 73 /* Work around OpenVMS incompatibility with C99. */ 74 #if !defined LLONG_MAX && defined __INT64_MAX 75 # define LLONG_MAX __INT64_MAX 76 # define LLONG_MIN __INT64_MIN 77 #endif 78 79 /* This include file assumes that signed types are two's complement without 80 padding bits; the above macros have undefined behavior otherwise. 81 If this is a problem for you, please let us know how to fix it for your host. 82 This assumption is tested by the intprops-tests module. */ 83 84 /* Does the __typeof__ keyword work? This could be done by 85 'configure', but for now it's easier to do it by hand. */ 86 #if (2 <= __GNUC__ \ 87 || (1210 <= __IBMC__ && defined __IBM__TYPEOF__) \ 88 || (0x5110 <= __SUNPRO_C && !__STDC__)) 89 # define _GL_HAVE___TYPEOF__ 1 90 #else 91 # define _GL_HAVE___TYPEOF__ 0 92 #endif 93 94 /* Return 1 if the integer type or expression T might be signed. Return 0 95 if it is definitely unsigned. This macro does not evaluate its argument, 96 and expands to an integer constant expression. */ 97 #if _GL_HAVE___TYPEOF__ 98 # define _GL_SIGNED_TYPE_OR_EXPR(t) TYPE_SIGNED (__typeof__ (t)) 99 #else 100 # define _GL_SIGNED_TYPE_OR_EXPR(t) 1 101 #endif 102 103 /* Bound on length of the string representing an unsigned integer 104 value representable in B bits. log10 (2.0) < 146/485. The 105 smallest value of B where this bound is not tight is 2621. */ 106 #define INT_BITS_STRLEN_BOUND(b) (((b) * 146 + 484) / 485) 107 108 /* Bound on length of the string representing an integer type or expression T. 109 Subtract 1 for the sign bit if T is signed, and then add 1 more for 110 a minus sign if needed. 111 112 Because _GL_SIGNED_TYPE_OR_EXPR sometimes returns 0 when its argument is 113 signed, this macro may overestimate the true bound by one byte when 114 applied to unsigned types of size 2, 4, 16, ... bytes. */ 115 #define INT_STRLEN_BOUND(t) \ 116 (INT_BITS_STRLEN_BOUND (TYPE_WIDTH (t) - _GL_SIGNED_TYPE_OR_EXPR (t)) \ 117 + _GL_SIGNED_TYPE_OR_EXPR (t)) 118 119 /* Bound on buffer size needed to represent an integer type or expression T, 120 including the terminating null. */ 121 #define INT_BUFSIZE_BOUND(t) (INT_STRLEN_BOUND (t) + 1) 122 123 124 /* Range overflow checks. 125 126 The INT_<op>_RANGE_OVERFLOW macros return 1 if the corresponding C 127 operators might not yield numerically correct answers due to 128 arithmetic overflow. They do not rely on undefined or 129 implementation-defined behavior. Their implementations are simple 130 and straightforward, but they are a bit harder to use than the 131 INT_<op>_OVERFLOW macros described below. 132 133 Example usage: 134 135 long int i = ...; 136 long int j = ...; 137 if (INT_MULTIPLY_RANGE_OVERFLOW (i, j, LONG_MIN, LONG_MAX)) 138 printf ("multiply would overflow"); 139 else 140 printf ("product is %ld", i * j); 141 142 Restrictions on *_RANGE_OVERFLOW macros: 143 144 These macros do not check for all possible numerical problems or 145 undefined or unspecified behavior: they do not check for division 146 by zero, for bad shift counts, or for shifting negative numbers. 147 148 These macros may evaluate their arguments zero or multiple times, 149 so the arguments should not have side effects. The arithmetic 150 arguments (including the MIN and MAX arguments) must be of the same 151 integer type after the usual arithmetic conversions, and the type 152 must have minimum value MIN and maximum MAX. Unsigned types should 153 use a zero MIN of the proper type. 154 155 These macros are tuned for constant MIN and MAX. For commutative 156 operations such as A + B, they are also tuned for constant B. */ 157 158 /* Return 1 if A + B would overflow in [MIN,MAX] arithmetic. 159 See above for restrictions. */ 160 #define INT_ADD_RANGE_OVERFLOW(a, b, min, max) \ 161 ((b) < 0 \ 162 ? (a) < (min) - (b) \ 163 : (max) - (b) < (a)) 164 165 /* Return 1 if A - B would overflow in [MIN,MAX] arithmetic. 166 See above for restrictions. */ 167 #define INT_SUBTRACT_RANGE_OVERFLOW(a, b, min, max) \ 168 ((b) < 0 \ 169 ? (max) + (b) < (a) \ 170 : (a) < (min) + (b)) 171 172 /* Return 1 if - A would overflow in [MIN,MAX] arithmetic. 173 See above for restrictions. */ 174 #define INT_NEGATE_RANGE_OVERFLOW(a, min, max) \ 175 ((min) < 0 \ 176 ? (a) < - (max) \ 177 : 0 < (a)) 178 179 /* Return 1 if A * B would overflow in [MIN,MAX] arithmetic. 180 See above for restrictions. Avoid && and || as they tickle 181 bugs in Sun C 5.11 2010/08/13 and other compilers; see 182 <https://lists.gnu.org/r/bug-gnulib/2011-05/msg00401.html>. */ 183 #define INT_MULTIPLY_RANGE_OVERFLOW(a, b, min, max) \ 184 ((b) < 0 \ 185 ? ((a) < 0 \ 186 ? (a) < (max) / (b) \ 187 : (b) == -1 \ 188 ? 0 \ 189 : (min) / (b) < (a)) \ 190 : (b) == 0 \ 191 ? 0 \ 192 : ((a) < 0 \ 193 ? (a) < (min) / (b) \ 194 : (max) / (b) < (a))) 195 196 /* Return 1 if A / B would overflow in [MIN,MAX] arithmetic. 197 See above for restrictions. Do not check for division by zero. */ 198 #define INT_DIVIDE_RANGE_OVERFLOW(a, b, min, max) \ 199 ((min) < 0 && (b) == -1 && (a) < - (max)) 200 201 /* Return 1 if A % B would overflow in [MIN,MAX] arithmetic. 202 See above for restrictions. Do not check for division by zero. 203 Mathematically, % should never overflow, but on x86-like hosts 204 INT_MIN % -1 traps, and the C standard permits this, so treat this 205 as an overflow too. */ 206 #define INT_REMAINDER_RANGE_OVERFLOW(a, b, min, max) \ 207 INT_DIVIDE_RANGE_OVERFLOW (a, b, min, max) 208 209 /* Return 1 if A << B would overflow in [MIN,MAX] arithmetic. 210 See above for restrictions. Here, MIN and MAX are for A only, and B need 211 not be of the same type as the other arguments. The C standard says that 212 behavior is undefined for shifts unless 0 <= B < wordwidth, and that when 213 A is negative then A << B has undefined behavior and A >> B has 214 implementation-defined behavior, but do not check these other 215 restrictions. */ 216 #define INT_LEFT_SHIFT_RANGE_OVERFLOW(a, b, min, max) \ 217 ((a) < 0 \ 218 ? (a) < (min) >> (b) \ 219 : (max) >> (b) < (a)) 220 221 /* True if __builtin_add_overflow (A, B, P) works when P is non-null. */ 222 #if 5 <= __GNUC__ && !defined __ICC 223 # define _GL_HAS_BUILTIN_OVERFLOW 1 224 #else 225 # define _GL_HAS_BUILTIN_OVERFLOW 0 226 #endif 227 228 /* True if __builtin_add_overflow_p (A, B, C) works. */ 229 #define _GL_HAS_BUILTIN_OVERFLOW_P (7 <= __GNUC__) 230 231 /* The _GL*_OVERFLOW macros have the same restrictions as the 232 *_RANGE_OVERFLOW macros, except that they do not assume that operands 233 (e.g., A and B) have the same type as MIN and MAX. Instead, they assume 234 that the result (e.g., A + B) has that type. */ 235 #if _GL_HAS_BUILTIN_OVERFLOW_P 236 # define _GL_ADD_OVERFLOW(a, b, min, max) \ 237 __builtin_add_overflow_p (a, b, (__typeof__ ((a) + (b))) 0) 238 # define _GL_SUBTRACT_OVERFLOW(a, b, min, max) \ 239 __builtin_sub_overflow_p (a, b, (__typeof__ ((a) - (b))) 0) 240 # define _GL_MULTIPLY_OVERFLOW(a, b, min, max) \ 241 __builtin_mul_overflow_p (a, b, (__typeof__ ((a) * (b))) 0) 242 #else 243 # define _GL_ADD_OVERFLOW(a, b, min, max) \ 244 ((min) < 0 ? INT_ADD_RANGE_OVERFLOW (a, b, min, max) \ 245 : (a) < 0 ? (b) <= (a) + (b) \ 246 : (b) < 0 ? (a) <= (a) + (b) \ 247 : (a) + (b) < (b)) 248 # define _GL_SUBTRACT_OVERFLOW(a, b, min, max) \ 249 ((min) < 0 ? INT_SUBTRACT_RANGE_OVERFLOW (a, b, min, max) \ 250 : (a) < 0 ? 1 \ 251 : (b) < 0 ? (a) - (b) <= (a) \ 252 : (a) < (b)) 253 # define _GL_MULTIPLY_OVERFLOW(a, b, min, max) \ 254 (((min) == 0 && (((a) < 0 && 0 < (b)) || ((b) < 0 && 0 < (a)))) \ 255 || INT_MULTIPLY_RANGE_OVERFLOW (a, b, min, max)) 256 #endif 257 #define _GL_DIVIDE_OVERFLOW(a, b, min, max) \ 258 ((min) < 0 ? (b) == _GL_INT_NEGATE_CONVERT (min, 1) && (a) < - (max) \ 259 : (a) < 0 ? (b) <= (a) + (b) - 1 \ 260 : (b) < 0 && (a) + (b) <= (a)) 261 #define _GL_REMAINDER_OVERFLOW(a, b, min, max) \ 262 ((min) < 0 ? (b) == _GL_INT_NEGATE_CONVERT (min, 1) && (a) < - (max) \ 263 : (a) < 0 ? (a) % (b) != ((max) - (b) + 1) % (b) \ 264 : (b) < 0 && ! _GL_UNSIGNED_NEG_MULTIPLE (a, b, max)) 265 266 /* Return a nonzero value if A is a mathematical multiple of B, where 267 A is unsigned, B is negative, and MAX is the maximum value of A's 268 type. A's type must be the same as (A % B)'s type. Normally (A % 269 -B == 0) suffices, but things get tricky if -B would overflow. */ 270 #define _GL_UNSIGNED_NEG_MULTIPLE(a, b, max) \ 271 (((b) < -_GL_SIGNED_INT_MAXIMUM (b) \ 272 ? (_GL_SIGNED_INT_MAXIMUM (b) == (max) \ 273 ? (a) \ 274 : (a) % (_GL_INT_CONVERT (a, _GL_SIGNED_INT_MAXIMUM (b)) + 1)) \ 275 : (a) % - (b)) \ 276 == 0) 277 278 /* Check for integer overflow, and report low order bits of answer. 279 280 The INT_<op>_OVERFLOW macros return 1 if the corresponding C operators 281 might not yield numerically correct answers due to arithmetic overflow. 282 The INT_<op>_WRAPV macros also store the low-order bits of the answer. 283 These macros work correctly on all known practical hosts, and do not rely 284 on undefined behavior due to signed arithmetic overflow. 285 286 Example usage, assuming A and B are long int: 287 288 if (INT_MULTIPLY_OVERFLOW (a, b)) 289 printf ("result would overflow\n"); 290 else 291 printf ("result is %ld (no overflow)\n", a * b); 292 293 Example usage with WRAPV flavor: 294 295 long int result; 296 bool overflow = INT_MULTIPLY_WRAPV (a, b, &result); 297 printf ("result is %ld (%s)\n", result, 298 overflow ? "after overflow" : "no overflow"); 299 300 Restrictions on these macros: 301 302 These macros do not check for all possible numerical problems or 303 undefined or unspecified behavior: they do not check for division 304 by zero, for bad shift counts, or for shifting negative numbers. 305 306 These macros may evaluate their arguments zero or multiple times, so the 307 arguments should not have side effects. 308 309 The WRAPV macros are not constant expressions. They support only 310 +, binary -, and *. The result type must be signed. 311 312 These macros are tuned for their last argument being a constant. 313 314 Return 1 if the integer expressions A * B, A - B, -A, A * B, A / B, 315 A % B, and A << B would overflow, respectively. */ 316 317 #define INT_ADD_OVERFLOW(a, b) \ 318 _GL_BINARY_OP_OVERFLOW (a, b, _GL_ADD_OVERFLOW) 319 #define INT_SUBTRACT_OVERFLOW(a, b) \ 320 _GL_BINARY_OP_OVERFLOW (a, b, _GL_SUBTRACT_OVERFLOW) 321 #if _GL_HAS_BUILTIN_OVERFLOW_P 322 # define INT_NEGATE_OVERFLOW(a) INT_SUBTRACT_OVERFLOW (0, a) 323 #else 324 # define INT_NEGATE_OVERFLOW(a) \ 325 INT_NEGATE_RANGE_OVERFLOW (a, _GL_INT_MINIMUM (a), _GL_INT_MAXIMUM (a)) 326 #endif 327 #define INT_MULTIPLY_OVERFLOW(a, b) \ 328 _GL_BINARY_OP_OVERFLOW (a, b, _GL_MULTIPLY_OVERFLOW) 329 #define INT_DIVIDE_OVERFLOW(a, b) \ 330 _GL_BINARY_OP_OVERFLOW (a, b, _GL_DIVIDE_OVERFLOW) 331 #define INT_REMAINDER_OVERFLOW(a, b) \ 332 _GL_BINARY_OP_OVERFLOW (a, b, _GL_REMAINDER_OVERFLOW) 333 #define INT_LEFT_SHIFT_OVERFLOW(a, b) \ 334 INT_LEFT_SHIFT_RANGE_OVERFLOW (a, b, \ 335 _GL_INT_MINIMUM (a), _GL_INT_MAXIMUM (a)) 336 337 /* Return 1 if the expression A <op> B would overflow, 338 where OP_RESULT_OVERFLOW (A, B, MIN, MAX) does the actual test, 339 assuming MIN and MAX are the minimum and maximum for the result type. 340 Arguments should be free of side effects. */ 341 #define _GL_BINARY_OP_OVERFLOW(a, b, op_result_overflow) \ 342 op_result_overflow (a, b, \ 343 _GL_INT_MINIMUM (0 * (b) + (a)), \ 344 _GL_INT_MAXIMUM (0 * (b) + (a))) 345 346 /* Store the low-order bits of A + B, A - B, A * B, respectively, into *R. 347 Return 1 if the result overflows. See above for restrictions. */ 348 #define INT_ADD_WRAPV(a, b, r) \ 349 _GL_INT_OP_WRAPV (a, b, r, +, __builtin_add_overflow, INT_ADD_OVERFLOW) 350 #define INT_SUBTRACT_WRAPV(a, b, r) \ 351 _GL_INT_OP_WRAPV (a, b, r, -, __builtin_sub_overflow, INT_SUBTRACT_OVERFLOW) 352 #define INT_MULTIPLY_WRAPV(a, b, r) \ 353 _GL_INT_OP_WRAPV (a, b, r, *, __builtin_mul_overflow, INT_MULTIPLY_OVERFLOW) 354 355 /* Nonzero if this compiler has GCC bug 68193 or Clang bug 25390. See: 356 https://gcc.gnu.org/bugzilla/show_bug.cgi?id=68193 357 https://llvm.org/bugs/show_bug.cgi?id=25390 358 For now, assume all versions of GCC-like compilers generate bogus 359 warnings for _Generic. This matters only for older compilers that 360 lack __builtin_add_overflow. */ 361 #if __GNUC__ 362 # define _GL__GENERIC_BOGUS 1 363 #else 364 # define _GL__GENERIC_BOGUS 0 365 #endif 366 367 /* Store the low-order bits of A <op> B into *R, where OP specifies 368 the operation. BUILTIN is the builtin operation, and OVERFLOW the 369 overflow predicate. Return 1 if the result overflows. See above 370 for restrictions. */ 371 #if _GL_HAS_BUILTIN_OVERFLOW 372 # define _GL_INT_OP_WRAPV(a, b, r, op, builtin, overflow) builtin (a, b, r) 373 #elif 201112 <= __STDC_VERSION__ && !_GL__GENERIC_BOGUS 374 # define _GL_INT_OP_WRAPV(a, b, r, op, builtin, overflow) \ 375 (_Generic \ 376 (*(r), \ 377 signed char: \ 378 _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned int, \ 379 signed char, SCHAR_MIN, SCHAR_MAX), \ 380 short int: \ 381 _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned int, \ 382 short int, SHRT_MIN, SHRT_MAX), \ 383 int: \ 384 _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned int, \ 385 int, INT_MIN, INT_MAX), \ 386 long int: \ 387 _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned long int, \ 388 long int, LONG_MIN, LONG_MAX), \ 389 long long int: \ 390 _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned long long int, \ 391 long long int, LLONG_MIN, LLONG_MAX))) 392 #else 393 # define _GL_INT_OP_WRAPV(a, b, r, op, builtin, overflow) \ 394 (sizeof *(r) == sizeof (signed char) \ 395 ? _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned int, \ 396 signed char, SCHAR_MIN, SCHAR_MAX) \ 397 : sizeof *(r) == sizeof (short int) \ 398 ? _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned int, \ 399 short int, SHRT_MIN, SHRT_MAX) \ 400 : sizeof *(r) == sizeof (int) \ 401 ? _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned int, \ 402 int, INT_MIN, INT_MAX) \ 403 : _GL_INT_OP_WRAPV_LONGISH(a, b, r, op, overflow)) 404 # ifdef LLONG_MAX 405 # define _GL_INT_OP_WRAPV_LONGISH(a, b, r, op, overflow) \ 406 (sizeof *(r) == sizeof (long int) \ 407 ? _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned long int, \ 408 long int, LONG_MIN, LONG_MAX) \ 409 : _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned long long int, \ 410 long long int, LLONG_MIN, LLONG_MAX)) 411 # else 412 # define _GL_INT_OP_WRAPV_LONGISH(a, b, r, op, overflow) \ 413 _GL_INT_OP_CALC (a, b, r, op, overflow, unsigned long int, \ 414 long int, LONG_MIN, LONG_MAX) 415 # endif 416 #endif 417 418 /* Store the low-order bits of A <op> B into *R, where the operation 419 is given by OP. Use the unsigned type UT for calculation to avoid 420 overflow problems. *R's type is T, with extrema TMIN and TMAX. 421 T must be a signed integer type. Return 1 if the result overflows. */ 422 #define _GL_INT_OP_CALC(a, b, r, op, overflow, ut, t, tmin, tmax) \ 423 (sizeof ((a) op (b)) < sizeof (t) \ 424 ? _GL_INT_OP_CALC1 ((t) (a), (t) (b), r, op, overflow, ut, t, tmin, tmax) \ 425 : _GL_INT_OP_CALC1 (a, b, r, op, overflow, ut, t, tmin, tmax)) 426 #define _GL_INT_OP_CALC1(a, b, r, op, overflow, ut, t, tmin, tmax) \ 427 ((overflow (a, b) \ 428 || (EXPR_SIGNED ((a) op (b)) && ((a) op (b)) < (tmin)) \ 429 || (tmax) < ((a) op (b))) \ 430 ? (*(r) = _GL_INT_OP_WRAPV_VIA_UNSIGNED (a, b, op, ut, t), 1) \ 431 : (*(r) = _GL_INT_OP_WRAPV_VIA_UNSIGNED (a, b, op, ut, t), 0)) 432 433 /* Return the low-order bits of A <op> B, where the operation is given 434 by OP. Use the unsigned type UT for calculation to avoid undefined 435 behavior on signed integer overflow, and convert the result to type T. 436 UT is at least as wide as T and is no narrower than unsigned int, 437 T is two's complement, and there is no padding or trap representations. 438 Assume that converting UT to T yields the low-order bits, as is 439 done in all known two's-complement C compilers. E.g., see: 440 https://gcc.gnu.org/onlinedocs/gcc/Integers-implementation.html 441 442 According to the C standard, converting UT to T yields an 443 implementation-defined result or signal for values outside T's 444 range. However, code that works around this theoretical problem 445 runs afoul of a compiler bug in Oracle Studio 12.3 x86. See: 446 https://lists.gnu.org/r/bug-gnulib/2017-04/msg00049.html 447 As the compiler bug is real, don't try to work around the 448 theoretical problem. */ 449 450 #define _GL_INT_OP_WRAPV_VIA_UNSIGNED(a, b, op, ut, t) \ 451 ((t) ((ut) (a) op (ut) (b))) 452 453 #endif /* _GL_INTPROPS_H */ 454